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Moments and cumulants of polynomial random variables on unitary groups, the Itzykson-Zuber integral, and free probability. (English) Zbl 1049.60091
Extending results of D. Weingarten [J. Math. Phys. 19, 999–1001 (1978; Zbl 0388.28013)], this paper gives explicit formulas for the Haar-average of polynomial functions on the unitary group \(U_d\). This is applied to probabilistic limit theorems for large \(d\), in particular to results of asymptotic freeness in quantum probability.

60K40 Other physical applications of random processes
81S25 Quantum stochastic calculus
60B15 Probability measures on groups or semigroups, Fourier transforms, factorization
43A75 Harmonic analysis on specific compact groups
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